In the course of deriving these expressions, Macdonald asks us to assume that $\phi$, which is the Sharpe ratio for the motion, is constant. But we can see that it is only constant when $a = 0$.

Later, Macdonald talks about the Sharpe ratio for the "interest rate risk", a phrasing I find very obscure. Is that the bond price process? The Vasicek process? In either case, they're driven by the same Brownian motion and should have the same (non-constant) Sharpe ratio.

Can somebody explain how to apply these formulas? Just a sketch would do -- but I'm stymied by the presentation.

1 Answer
1

Bond Prices

Assume that the short rate $r_t$ follows the Ito process as described by the following stochastic differential equation
\begin{align}
d{{r}_{t}}=\mu ({{r}_{t}},t)dt+\sigma ({{r}_{t}},t)d{{W}_{t}^{P}}
\end{align}
we assume the bond price to be dependent on $r_t$ only, independent of default risk, liquidity and other factors. If we write the bond price as $P(r_t, t)=V(t,r_t,T)$ such that $V(t,r_t,t)=1$ then
\begin{align}
dV=({{V}_{t}}+\mu \,{{V}_{r}}\,+\frac{1}{2}{{\sigma }^{2}}{{V}_{rr}})dt+\sigma {{V}_{r}}d{{W}_{t}}
\end{align}
for simplicity let
\begin{align}
& {{\mu }_{V}}=\frac{{{V}_{t}}+\mu {{V}_{r}}\,+\frac{1}{2}{{\sigma }^{2}}{{V}_{rr}}}{V} \\
& {{\sigma }_{V}}=\frac{\sigma {{V}_{r}}\,}{V} \\
\end{align}
thus we have
\begin{align}
dV={{\mu }_{V}}\,Vdt+{{\sigma }_{V}}\,V\,d{{W}_{t}}
\end{align}
The following portfolio is constructed: we buy a bond of dollar value V1 with maturity $T_1$ and sell another bond of dollar
value $V_2$ with maturity $T_2$. The portfolio value $\Pi$ is given by
\begin{align}
\Pi ={{V}_{1}}-{{V}_{2}}
\end{align}
According to the bond price dynamics,we have
\begin{align}
\Pi =({{\mu }_{{{V}_{1}}}}{{V}_{1}}-{{\mu }_{{{V}_{2}}}}{{V}_{2}})\,dt+({{\sigma }_{{{V}_{1}}}}{{V}_{1}}-{{\sigma }_{{{V}_{2}}}}{{V}_{2}})\,d{{W}_{t}}
\end{align}
Suppose $V_1$ and $V_2$ are chosen such that
\begin{align}
& {{V}_{1}}=\frac{{{\sigma }_{{{V}_{2}}}}}{{{\sigma }_{{{V}_{2}}}}-{{\sigma }_{{{V}_{1}}}}}\Pi \\
& {{V}_{2}}=\frac{{{\sigma }_{{{V}_{1}}}}}{{{\sigma }_{{{V}_{2}}}}-{{\sigma }_{{{V}_{1}}}}}\Pi \\
\end{align}
then the stochastic term in $d\Pi$ vanishes and the equation becomes
$$d\Pi =\left( \frac{{{\mu }_{{{V}_{1}}}}{{\sigma }_{{{V}_{2}}}}-{{\mu }_{{{V}_{2}}}}{{\sigma }_{{{V}_{1}}}}}{{{\sigma }_{{{V}_{2}}}}-{{\sigma }_{{{V}_{1}}}}} \right)\Pi \,dt$$
Since the portfolio is instantaneously riskless, in order to avoid arbitrage opportunities,it must earn the riskless short rate so that $d\Pi =r(t)\Pi dt$ ,then
$$\frac{{{\mu }_{{{V}_{1}}}}-r(t)}{{{\sigma }_{{{V}_{1}}}}}=\frac{{{\mu }_{{{V}_{2}}}}-r(t)}{{{\sigma }_{{{V}_{2}}}}}$$
The above relation is valid for arbitrary maturity dates $T_1$ and $T_2$, so the ratio should be independent of maturity $T$.Let the common ratio be defined
by $\lambda$, that is,
$$\frac{{{\mu }_{V}}-r(t)}{{{\sigma }_{V}}}=\lambda \,({{r}_{t}},t)$$
The quantity $\lambda$ is called the market price of risk of the short rate.If we substitute $μ_V(r, t)$ and $σ_V(r, t)$ into above Equation, we obtain the following governing differential equation for the price of a zero-coupon bond
$${{V}_{t}}+(\mu -\lambda \sigma \,){{V}_{r}}\,+\frac{1}{2}{{\sigma }^{2}}{{V}_{rr}}-{{r}_{t}}\,V=0$$

Change Measure

we assume $Q$ be a martingale measure such that
$$dW_{t}^{P}=-\lambda(r,t)dt+dW_{t}^{Q}$$
thus we have
$$d{{r}_{t}}=\mu^*(r_t,t)dt+\sigma ({{r}_{t}},t)dW_{t}^{Q}$$
where
$$\mu^*(r_t,t)=\mu({{r}_{t}},t)-\lambda ({{r}_{t}},t)\sigma ({{r}_{t}},t)$$

Affine Term Structure Models

A short rate model that generates the bond price solution of the form
$$P(t\,,T)=V(t,r_t,T)={{e}^{A(t,T)\,-\,B(t,T){{r}_{t}}\,}}$$
Suppose the dynamics of the short rate $r_t$ under the risk neutral measure $Q$ is governed by
\begin{align}
d{{r}_{t}}=\mu^* ({{r}_{t}},t)dt+\sigma ({{r}_{t}},t)d{{W}_{t}^{Q}}
\end{align}
where
\begin{align}
&\mu^* ({{r}_{t}},T)=\alpha (t)\,{{r}_{t}}+\beta (t) \\
&{{\sigma }^{2}}({{r}_{t}},T)=\gamma (t)\,{{r}_{t}}+\delta (t) \\
\end{align}
We show the governing equation for $P(t, T )=V(t,r,T)$ is given by
$${{V}_{t}}+\mu ^*{{V}_{r}}\,+\frac{1}{2}{{\sigma }^{2}}{{V}_{rr}}-{{r}_{t}}\,V=0$$